Let's examine f(x1,x2,x3,x4)=x1*x2+x3*x4.
It is D4-invariant. I was able to calculate that it has 3 possible
outcomes out of 4!=24 arrangements of arguments:
var('x1, x2, x3, x4')
lista = [x1, x2, x3, x4]
def f(y):
return lista[y[0]]*lista[y[1]]+lista[y[2]]*lista[y[3]]
list(set(map(f, arrangements([0,1,2,3], 4))))
--> [x1*x4 + x2*x3, x1*x3 + x2*x4, x1*x2 + x3*x4]
However, this only works because Sage happens to arrange monomials. How to
compare polynomials? This does not work:
(x1*x2+x3*x4)==(x2*x1+x3*x4)
So how to tell Sage that I am comparing polynomials, and not for example
matrices?
And a harder question: how to show that f really is D4-invariant, not
just G-invariant where |G|=4!/3? Or, in general, calculate "invariantness"
of multivariate polynomial?
--
Jori Mäntysalo
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